TIFR 2013 QUESTION PPR

7/27/2019 TIFR 2013 QUESTION PPR

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GS-2013

(Computer & Systems Sciences)

TATA INSTITUTE OF FUNDAMENTAL RESEARCH

Written Test in COMPUTER & SYSTEMS SCIENCES - December 9, 2012

Duration : Three hours (3 hours)

Name : _______________________________________________ Ref. Code : ____________

Please read all instructions carefully before you attempt the questions.

1. Please fill-in details about name, reference code etc. on the answer sheet. The Answer

Sheet is machine readable. Read the instructions given on the reverse of the answer sheet

before you start filling it up. Use only HB pencils to fill-in the answer sheet.

2. Indicate your ANSWER ON THE ANSWER SHEET by blackening the appropriate circle for

each question. Do not mark more than one circle for any question : this will be treated as a

wrong answer.

3. This question paper consists of three (3) parts. Part-A contains twenty (20) questions and

must be attempted by all candidates. Part-B & Part-C contain twenty (20) questions

each, directed towards candidates for (B) Computer Science and (C) Systems Science

(including Communications & Math Finance), respectively. STUDENTS MAY ATTEMPT

EITHER PART-B OR PART-C. In case, a student attempts both Parts B & C (no extra time

will be given) and qualifies for interview in both B & C, he/she will have opportunity to

be interviewed in both areas. All questions carry equal marks. A correct answer for a

question will give you +4 marks, a wrong answer will give you -1 mark, and a question not

answered will not get you any marks.

4. We advise you to first mark the correct answers in the QUESTION SHEET and then to

TRANSFER these to the ANSWER SHEET only when you are sure of your choice.

5. Rough work may be done on blank pages of the question paper. If needed, you may ask for

extra rough sheets from an Invigilator.

6. Use of calculators is NOT permitted.

7. Do NOT ask for clarifications from the invigilators regarding the questions. They have been

instructed not to respond to any such inquiries from candidates. In case a

correction/clarification is deemed necessary, the invigilator(s) will announce it publicly.

CSS

Correct answers are ticked in green.


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Part A

Common Questions

1. An infinite two-dimensional pattern is indicated below.

e

eeeeeeeeeee

e

eeeeeeeeeee

eeeeeeeeeeee

eeeeeeeeeeee

e

eeeeeeeeeee

e

eeeeeeeeeee

e

eeeeeeeeeee

e

eeeeeeeeeee

The smallest closed figure made by the lines is called a unit triangle. Within every unit triangle, there isa mouse. At every vertex there is a laddoo. What is the average number ofladdoos per mouse?

(a) 3

(b) 2

(c) 1

(d) 12

(e) 13

2. Consider the following two types of elections to determine which of two parties A and B forms the

next government in the 2014 Indian elections. Assume for simplicity an Indian population of size 545545(= 545*1001). There are only two parties A and B and every citizen votes.

TYPE C: The country is divided into 545 constituencies and each constituency has 1001 voters. Elec-

tions are held for each constituency and a party is said to win a constituency if it receive a majority of the

vote in that constituency. The party that wins the most constituencies forms the next government.

TYPE P: There are no constituencies in this model. Elections are held throughout the country and the

party that wins the most votes (among 545545 voters) forms the government.

Which of the following is true?

(a) If a party forms the govt. by election TYPE C winning at least two-thirds of the constituencies, then

it will also form the govt. by election TYPE P.

(b) If a party forms the govt. by election TYPE C, then it will also form the govt. by election TYPE P.

(c) If a party forms the govt. by election TYPE P, then it will also form the govt. by election TYPE C.

(d) All of the above

(e) None of the above

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3. Three candidates, Amar, Birendra and Chanchal stand for the local election. Opinion polls are

conducted and show that fraction a of the voters prefer Amar to Birendra, fraction b prefer Birendra toChanchal and fraction c prefer Chanchal to Amar. Which of the following is impossible?

(a) (a,b,c) = (0.51, 0.51, 0.51);

(b) (a,b,c) = (0.61, 0.71, 0.67);

(c) (a,b,c) = (0.68, 0.68, 0.68);

(d) (a,b,c) = (0.49, 0.49, 0.49);

(e) None of the above.

4. A biased coin is tossed repeatedly. Assume that the outcomes of different tosses are independent and

probability of heads is 2/3 in each toss. What is the probability of obtaining an even number of heads in 5tosses, zero being treated as an even number?

(a) 121/243

(b) 122/243

(c) 124/243

(d) 125/243

(e) 128/243

5. The late painter Maqbool Fida Husain once coloured the surface of a huge hollow steel sphere, of

radius 1 metre, using just two colours, red and blue. As was his style however, both the red and blue areas

were a bunch of highly irregular disconnected regions. The late sculptor Ramkinkar Baij then tried to fit in

a cube inside the sphere, the eight vertices of the cube touching only red coloured parts of the surface of the

sphere. Assume = 3.14 for solving this problem. Which of the following is true?

(a) Baij is bound to succeed if the area of the red part is 10 sq. metres;

(b) Baij is bound to fail if the area of the red part is 10 sq. metres;

(c) Baij is bound to fail if the area of the red part is 11 sq. metres;

(d) Baij is bound to succeed if the area of the red part is 11 sq. metres;

(e) None of the above.

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6. You are lost in the national park of Kabrastan. The park population consists of tourists and kabras-

tanis. Tourists comprise two-thirds of the population the park, and give a correct answer to requests for

directions with probability 3/4. The air of Kabrastan has an amnesaic quality however, and so the answers

to repeated questions to tourists are independent, even if the question and the person are the same. If youask a Kabrastani for directions, the answer is always wrong.

Suppose you ask a randomly chosen passer-by whether the exit from the park is East or West. The

answer is East. You then ask the same person again, and the reply is again East. What is the probability of

East being correct?

(a) 1/4

(b) 1/3

(c) 1/2

(d) 2/3

(e) 3/4

7. For any complex number z, arg z defines its phase, chosen to be in the interval 0 arg z < 360o. Ifz1, z2 and z3 are three complex numbers with the same modulus but different phases (arg z3 < arg z2 < argz1 < 180o), then the quantity

arg (z1/z2)

arg [(z1 z3)/(z2 z3)]is a constant, and has the value

(a) 2

(b) 13

(c) 1

(d) 3

(e) 12

8. Find the sum of the infinite series

1

1 3 5 +1

3 5 7 +1

5 7 9 +1

7 9 11 + .......

(a)

(b)

1

2

(c) 16

(d) 112

(e) 114

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9. There are n kingdoms and 2n champions. Each kingdom gets 2 champions. The number of ways inwhich this can be done is:

(a)(2n)!

2n

(b)(2n)!

n!

(c)(2n)!

2n n!(d) n!/2


10. Three men and three rakhsasas arrive together at a ferry crossing to find a boat with an oar, but no

boatman. The boat can carry one or at the most two persons, for example, one man and one rakhsasa, and

each man or rakhsasa can row. But if at any time, on any bank, (including those who maybe are in the boat

as it touches the bank) rakhsasas outnumber men, the former will eat up the latter. If all have to go to the

other side without any mishap, what is the minimum number of times that the boat must cross the river?

(a) 7

(b) 9

(c) 11

(d) 13

(e) 15

11. Let there be a pack of 100 cards numbered 1 to 100. The ith card states: There are at most i 1 truecards in this pack. Then, how many cards of the pack contain TRUE statements?

(a) 0

(b) 1

(c) 100

(d) 50


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12. Among numbers 1 to 1000 how many are divisible by 3 or 7?

(a) 333

(b) 142

(c) 475

(d) 428


13. Doctors A and B perform surgery on patients in stages III and IV of a disease. Doctor A has performed

a 100 surgeries (on 80 stage III and 20 stage IV patients) and 80 out of her 100 patients have survived (78

stage III and 2 stage IV survivors). Doctor B has also performed 100 surgeries (on 50 stage III and 50

stage IV patients). Her success rate is 60/100 (49 stage III survivors and 11 stage IV survivors). A patient

has been advised that she is equally likely to be suffering from stage III or stage IV of this disease. Which

doctor would you recommend to this patient and why?

(a) Doctor A since she has a higher success rate

(b) Doctor A since she specializes in stage III patients and the success of surgery in stage IV patients is

anyway too low

(c) Doctor B since she has performed more stage IV surgeries

(d) Doctor B since she appears to be more successful

(e) There is not enough data since the choice depends on the stage of the disease the patient is suffering

from

14. An unbiased die is thrown n times. The probability that the product of numbers would be even is

(a) 1/(2n)

(b) 1/[(6n)!]

(c) 1 6n

(d) 6n


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15. Let sgn(x) =

+1 ifx 01 ifx < 0 .

What is the value of the summation

50i=0

sgn( (2i 1)(2i 3) (2i 99) )

(a) 0

(b) 1(c) +1

(d) 25

(e) 50

16. The minimum of the function f(x) = x loge(x) over the interval [1/2, ) is(a) 0

(b) -e

(c) loge(2)/2(d) 1/e(e) None of the above

17. A stick of unit length is broken into two at a point chosen at random. Then, the larger part of the

stick is further divided into two parts in the ratio 4:3. What is the probability that the three sticks that are

left CANNOT form a triangle?

(a) 1/4

(b) 1/3

(c) 5/6

(d) 1/2

(e) loge(2)/2.

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18. Consider three independent uniformly distributed (taking values between 0 and 1) random variables.

What is the probability that the middle of the three values (between the lowest and the highest value) lies

between a and b where 0

a < b

1.

(a) 3(1 b)a(b a)(b) 3((b a) (b2 a2)/2)(c) 6(1 b)a(b a)(d) (1 b)a(b a)(e) 6((b2 a2)/2 (b3 a3)/3).

19. Consider a sequence of numbers (n : n = 1, 2, . . .), such that 1 = 10 and

n+1 =20n

20 + n

for n 1. Which of the following statements is true?Hint: Consider the sequence of reciprocals.

(a) The sequence (n : n = 1, 2, . . .) converges to zero.

(b) n 1 for all n.(c) The sequence (n : n = 1, 2, . . .) is decreasing and converges to 1.

(d) The sequence (n : n = 1, 2, . . .) is decreasing and then increasing. Finally it converges to 1.


20. Consider a well functioning clock where the hour, minute and the seconds needles are exactly at zero.

How much time later will the minutes needle be exactly one minute ahead (1/60 th of the circumference) of

the hours needle and the seconds needle again exactly at zero?

Hint: When the desired event happens both the hour needle and the minute needle have moved an integer

multiple of 1/60 th of the circumference.

(a) 144 minutes

(b) 66 minutes

(c) 96 minutes

(d) 72 minutes

(e) 132 minutes

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Part B

Computer Science Questions

1. Let G = (V, E) be a simple undirected graph on n vertices. A colouring of G is an assignmentof colours to each vertex such that endpoints of every edge are given different colours. Let (G) denotethe chromatic number ofG, i.e., the minimum number of colours needed for a valid colouring of G. A setB V is an independent set if no pair of vertices in B is connected by an edge. Let a(G) be the numberof vertices in a largest possible independent set in G. In the absence of any further information about G wecan conclude

(a) (G) a(G)(b) (G) a(G)(c) a(G) n/(G)

(d) a(G) n/(G)(e) None of the above

2. Consider polynomials in a single variable x of degree d. Suppose d < n/2. For such a polynomialp(x), let Cp denote the n-tuple (p(i))1in. For any two such distinct polynomials p, q, the number ofcoordinates where the tuples Cp, Cq differ is

(a) at most d

(b) at most n d(c) between d and n d

(d) at least n d(e) none of the above

3. How many 4 4 matrices with entries from {0, 1} have odd determinant?Hint: Use modulo 2 arithmetic.

(a) 20160

(b) 32767

(c) 49152

(d) 57343

(e) 65520

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4. A set S together with partial order is called a well order if it has no infinite descending chains, i.e.there is no infinite sequence x1, x2, . . . of elements from S such that xi+1 xi and xi+1 = xi for all i.Consider the set of all words (finite sequences of letters a-z), denoted by W, in dictionary order.

(a) Between aa and az there are only 24 words

(b) Between aa and az there are only 224 words

(c) W is not a partial order

(d) W is a partial order but not a well order

(e) W is a well order

5. Given a weighted directed graph with n vertices where edge weights are integers (positive, zero, ornegative), determining whether there are paths of arbitrarily large weight can be performed in time

(a) O(n)

(b) O(n log(n)) but not O(n)(c) O(n1.5) but not O(n log n)

(d) O(n3) but not O(n1.5)

(e) O(2n) but not O(n3)

6. Let L and L be languages over the alphabet . The left quotient ofL by L is

L/Ldef= {w : wx L for some x L}.

Which of the following is true?

(a) IfL/L is regular then L is regular

(b) IfL is regular then L/L is regular

(c) IfL/L is regular then L is regular

(d) L/L is a subset ofL

(e) IfL/L and L are regular, then L is regular

7. Which of the following is not implied by P = NP?

(a) 3SAT can be solved in polynomial time

(b) Halting problem can be solved in polynomial time

(c) Factoring can be solved in polynomial time.

(d) Graph isomorphism can be solved in polynomial time.

(e) Travelling salesman problem can be solved in polynomial time.

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8. Which one of the following languages over the alphabet {0, 1} is regular?(a) The language of balanced parentheses where 0, 1 are thought of as (, ) respectively

(b) The language of palindromes, i.e., bit strings x that read the same from left to right as well as right toleft

(c) L = {0m2 : 3 m}(d) The Kleene closure L, where L is the language in (c) above

(e) {0m1n | 1 m n}9. Suppose n straight lines are drawn on a plane. When these lines are removed, the plane falls apartinto several connected components called regions. A region R is said to be convex if it has the followingproperty: whenever two points are in R, then the entire line segment joining them is in R. Suppose no twoof the n lines are parallel. Which of the following is true?

(a) O(n) regions are produced, and each region is convex

(b) O(n2) regions are produced but they need not all be convex

(c) O(n2) regions are produced, and each region is convex

(d) O(n log n) regions are produced, but they need not all be convex

(e) All regions are convex but there may be exponentially many of them

10. Let m, n be positive integers with m a power of 2. Let s = 100n2 log m. Suppose S1, S2, . . . , S mare subsets of{1, 2, . . . , s} such that |Si| = 10n log m and |Si Sj | log m for all 1 i < j m. Sucha collection of sets S1, . . . , S

mis an example of a so-called Nisan-Wigderson design. We now consider the

set membership problem, where we have to store an arbitrary subset T {1, 2, . . . , m}, |T| = n as an arrayA of s bits so that given any integer x, 1 x m, we can discover whether x T by reading only onebit of A. Consider the following strategy to solve this problem. Array A is initialised to all zeroes. Giventhe set T to be stored, we put a one in all the locations of A indexed by the union tTSt. Now, given theinteger x, we read a random location in A from Sx and declare that x T iff the bit in that location is one.This strategy gives the correct answer with probability

(a) 1 ifx T and at most 0.1 ifx T.(b) at least 0.9 ifx T and at most 0.1 ifx T.(c) at least 0.9 ifx T and at least 0.9 ifx T.(d) 1 ifx T and at least 0.9 ifx T.(e) at least 0.9 ifx T and 1 ifx T.

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11. Which of the following statements is FALSE?

(a) The intersection of a context free language with a regular language is context free.

(b) The intersection of two regular languages is regular.

(c) The intersection of two context free languages is context free

(d) The intersection of a context free language and the complement of a regular language is context free.

(e) The intersection of a regular language and the complement of a regular language is regular.

12. It takes O(n) time to find the median in a list ofn elements, which are not necessarily in sorted orderwhile it takes only O(1) time to find the median in a list of n sorted elements. How much time does it taketo find the median of2n elements which are given as two lists ofn sorted elements each?

(a) O(1)

(b) O(log n) but not O(1)

(c) O(

n) but not O(log n)

(d) O(n) but not O(

n)

(e) O(n log n) but not O(n)

13. Given a binary tree of the following form and having n nodes, the height of the tree is

g i d i i

iiii

i g i

ed

g

eee

eeeeee

eeeeeeeee

eee

eeeeeeeee

(a) (log n)

(b) (n)

(c) (

n)

(d) (n/ log n)


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14. Assume a demand paged memory system where ONLY THREE pages can reside in the memory at a

time. The following sequence gives the order in which the program references the pages.

1, 3, 1, 3, 4, 2, 2, 4

Assume that least frequently used page is replaced when necessary. If there is more than one least frequently

used pages then the least recently used page among them is replaced. During the programs execution, how

many times will the pages 1,2,3 and 4 be brought to the memory?

(a) 2,2,2,2 times, respectively

(b) 1,1,1,2 times, respectively

(c) 1,1,1,1 times, respectively

(d) 2,1,2,2 times, respectively


15. Let G be an undirected graph with n vertices. For any subset S of vertices, the set of neighboursofS consists of the union ofS and the set of vertices S that are connected to some vertex in S by an edgeof G. The graph G has the nice property that every subset of vertices S of size at most n/2 has at least

1.5|S|-many neighbours. What is the length of a longest path in G?(a) O(1)

(b) O(log log n) but not O(1)

(c) O(log n) but not O(log log n)

(d) O(n) but not O(log n)(e) O(n) but not O(

n)

16. Consider a function Tk,n : {0, 1}n {0, 1} which returns 1 iff at least k of its n inputs are 1.Formally, Tk,n(x) = 1 iff

n1

xi k. Let y {0, 1}n be such that y has exactly k ones. Then, thefunction Tk,n1(y1, y2, . . . , yi1, yi+1, . . . , yn) (where yi is omitted) is equivalent to

(a) Tk1,n(y)

(b) Tk,n(y)

(c) yi

(d) yi(e) None of the above

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17. In a connected weighted graph with n vertices, all the edges have distinct positive integer weights.Then, the maximum number of minimum weight spanning trees in the graph is

(a) 1(b) n

(c) equal to number of edges in the graph

(d) equal to maximum weight of an edge of the graph

(e) nn2

18. Let S be a set of numbers. For x S, the rank of x is the number of elements in S that are lessthan or equal to x. The procedure Select(S, r) takes a set S of numbers and a rank r (1 r |S|) andreturns the element in S of rankr. The procedure MultiSelect(S, R) takes a set of numbers S and a list ofranks R =

{r1 < r2 0 for each n.

(c)

n=1 xn = and xn 0.01 infinitely often.(d)

n=1 xn = and each xn 1/n2.

(e)

n=1 xn < and each xn > xn+1.

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16. A surprise quiz contains three multiple choice questions; question 1 has 3 suggested answers,

question 2 has four, and question 3 has two. A completely unprepared student decides to choose the answers

at random. IfX is the number of questions the student answers correctly, the expected number of correct

answers is

(a) 15/12

(b) 7/12

(c) 13/12

(d) 18/12


17. Consider four coins, three of which are fair, that is they have heads on one side and tails on the

other and both are equally likely to occur in a toss. The fourth coin has heads on both sides. Given that one

coin amongst the four is picked at random and is tossed, and the outcome is seen to be head, what is the

probability that its other side is tails?

(a) 1/2

(b) 3/8

(c) 3/5

(d) 3/4

(e) 5/7

18. Consider a coin tossing game between Santa and Banta. Both of them toss two coins sequentially, first

Santa tosses a coin then Banta and so on. Santa tosses a fair coin: Probability of heads is 1/2 and probability

of tails is 1/2. Bantas coin probabilities depend on the outcome of Santas toss. If Santa sees an outcome

heads then Banta gets a coin whose probability of heads is 3/4, and of tails is 1/4. If Santa tosses a tail then

Banta gets a coin whose probability of tossing a tails is 3/4 and of heads is 1/4.

What is the probability of the event that they both have one head and one tail in the two trials conducted

by each of them?

(a) 1/2

(b) 5/16

(c) 3/16

(d) 1/4

(e) 1/3

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19. Which of the following is true for polynomials defined over real numbers R.

(a) Every odd degree polynomial has a real root.

(b) Every odd degree polynomial has at least one complex root.

(c) Every even degree polynomial has at least one complex root.

(d) Every even degree polynomial has a real root.


20. A function f : R R is convex if for x, y R, [0, 1], f(x+(1)y) f(x)+(1)f(y).Which of the following is not convex:

(a) x2

(b) x

3

(c) x

(d) x4

(e) ex

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TIFR 2013 QUESTION PPR

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